Consider a sesquilinear form \( t [\cdot ,\cdot ]\)  (see [1]} )
defined on a linear manifold of the Hilbert space \(\mathfrak {H}.\)  Denote by \( t [\cdot ]\)  the quadratic form corresponding to the sesquilinear form \( t [\cdot ,\cdot ].\) 
Let \( \mathfrak {h}=( t + t ^{\ast })/2,\, \mathfrak {k} =( t - t ^{\ast })/2i\) 
be a real and imaginary component of the form \( t \)  respectively, where \( t^{\ast }[u,v]=t \overline{[v,u]},\;\mathrm {D}(t ^{\ast })=\mathrm {D}(t).\)  According to these definitions, we have \(\mathfrak {h}[\cdot ]=\mathrm {Re}\,t[\cdot ],\, \mathfrak {k}[\cdot ]=\mathrm {Im}\,t[\cdot ].\)  Denote by \(\tilde{t}\)  the closure of a form \(t.\)  The range of a quadratic form
\( t [f],\,f\in \mathrm {D}(t),\,\Vert f\Vert _{\mathfrak {H}}=1\)  is called range of the sesquilinear form \(t \)  and is denoted by \(\Theta (t).\) 
A form \(t\)  is called sectorial if its range belongs to a sector having a vertex \(\gamma \)  situated at the real axis and a semi-angle \(0\le \theta <\pi /2.\)  Suppose \(t\)  is a closed sectorial form; then a linear manifold \(\mathrm {D}_{0}(t) \subset \mathrm {D} (t)\)  is
called core of \(t,\)  if the restriction of \(t\)  to \(\mathrm {D}_{0}(t)\)  has the closure
\(t\)  (see[1]}). Due to Theorem 2.7 [1]} there exist unique m-sectorial operators \(T_{t},T_{ \mathfrak {h}} \)  associated with the closed sectorial forms \(t, \mathfrak {h}\)  respectively. The operator \(T_{ \mathfrak {h}} \)  is called a real part of the operator \(T_{t}\)  and is denoted by \(Re\, T_{t}.\)  Suppose \(L\)  is a sectorial densely defined operator and \(t[u,v]:=(Lu,v)_{\mathfrak {H}},\,\mathrm {D}(t)=\mathrm {D}(L);\)  then
due to Theorem 1.27 [1]} the corresponding form \(t\)  is closable, due to
Theorem 2.7 [1]} there exists a unique m-sectorial operator \(T_{\tilde{t}}\)  associated with the form \(\tilde{t}.\)  In accordance with the definition [1]} the operator \(T_{\tilde{t}}\)  is called a Friedrichs extension of the operator \(L.\) 
Everywhere further, unless otherwise stated, we use notations of the papers [7]}, [1]}, [9]}, [10]},
[11]}.
